math 532 – homework set 3
february 24, 2016
1)
(a) Let 𝑓 ∶ 𝑋 → 𝑆 and 𝑔 ∶ 𝑌 → 𝑆 be morphisms of quasi-projective algebraic sets.
Describe (without proof) in more common terms the fiber product 𝑋 ×u� 𝑌 in the
following cases:
(A) 𝑆 = 𝔸0 = pt
(B) 𝑋 = 𝔸0 = pt
(C) 𝑓 is an immersion.
(D) 𝑓 and 𝑔 are immersions.
(b) Let 𝑋 → 𝑇 → 𝑆 and 𝑌 → 𝑆 be morphisms. Prove that
𝑋 ×u� (𝑇 ×u� 𝑌 ) ≅ 𝑋 ×u� 𝑌 .
(c) Let 𝑓 ∶ 𝑋 → 𝑌 and 𝑔 ∶ 𝑌 → 𝑍 be proper morphisms. Prove that 𝑔 ∘ 𝑓 is proper.
2) Let Grass(2, 4) ⊂ ℙ(⋀2 𝑘 4 ) = ℙ5 be the Grassmannian of 2-planes in 𝑘 4 . Let 𝑒u� be the
standard basis of 𝑘 4 and 𝑒u� ∧ 𝑒u� (1 ≤ 𝑖 < 𝑗 ≤ 4) be the induced basis of ⋀2 𝑘 4 . Finally
write 𝑥u�,u� for the corresponding homogeneous coordinates of ℙ(⋀2 𝑘 4 ) = ℙ5 (called
Plücker coordinates).
Show that Grass(2, 4) = 𝑍̃(𝑥1,2 𝑥3,4 − 𝑥1,3 𝑥2,4 + 𝑥1,4 𝑥2,3 ) ⊆ ℙ5 .
(It might be useful to read Chapter 8 of [g] to gain more familiarity with the exterior
product.)
3) Locate the singular points of the following surfaces in 𝔸3 (assume char 𝑘 ≠ 2). Which
is which in Figure 1?
(a) 𝑥𝑦2 = 𝑧2 ;
(b) 𝑥 2 + 𝑦2 = 𝑧2 ;
(c) 𝑥𝑦 + 𝑥 3 + 𝑦3 = 0.
4) Let 𝑃 be a point of an irreducible algebraic set 𝑋 and let 𝔪 be the maximal ideal of the
local ring 𝒪u�,u� . The Zariski tangent space 𝑇u� (𝑋) of 𝑋 at 𝑃 is the dual 𝑘 -vector space
of 𝔪/𝔪2 .
(a) Show that dim 𝑇u� (𝑋) ≥ dim 𝑋 with equality if and only if 𝑃 is non-singular.
Figure 1: Surface singularities (cf. [h, p. 36])
(b) Show that for any morphism 𝑓 ∶ 𝑋 → 𝑌 there is a natural induced 𝑘 -linear map
𝑇u� (𝑓 ) ∶ 𝑇u� (𝑋) → 𝑇u� (u�) (𝑌 ).
5)
(c) If 𝑓 is the vertical projection of the parabola 𝑥 = 𝑦2 onto the 𝑥 -axis, show that the
induced map 𝑇0 (𝑓 ) of tangent spaces at the origin is the zero map.
(a) Show that if 𝑓 ∶ ℙ1 − → ℙ1 is a rational map, then there exists a unique extension
of 𝑓 to a morphism ℙ1 → ℙ1 .
(b) Think of ℙ1 as 𝔸1 ∪ {∞}. Then we define a fractional linear transformation of ℙ1
by sending 𝑥 ↦ u�u�+u�
for 𝑎, 𝑏, 𝑐, 𝑑 ∈ 𝑘 with 𝑎𝑑 −𝑏𝑐 ≠ 0. Show that a fractional linear
u�u�+u�
transformation induces an automorphism of ℙ1 . Further show that the fractional
linear transformations form a group isomorphic to PGL2 (𝑘) = GL2 (𝑘)/𝑘 ∗ .
(c) Let Aut ℙ1 denote the group of all automorphisms of ℙ1 . Show that Aut ℙ1 ≅
Aut 𝑘(𝑥), the group of all 𝑘 -algebra automorphisms of the field 𝑘(𝑥).
(d) Now show that every automorphism of 𝑘(𝑥) is a fractional linear transformation,
and deduce that PGL2 (𝑘) → Aut ℙ1 is an isomorphism.
references
[g]
Andreas Gathmann. Algebraic Geometry. Class Notes TU Kaiserslautern 2014. url:
[h]
Robin Hartshorne. Algebraic geometry. Graduate Texts in Mathematics 52. New York:
Springer-Verlag, 1977. xvi+496. isbn: 0-387-90244-9.
http://www.mathematik.uni-kl.de/~gathmann/class/alggeom2014/main.pdf.